During the halftime of a soccer game, a sling shot launches T-shirts at the crowd. A T-shirt is launched from a height of 6 feet with an initial upward velocity of 64 feet per second. Use the equation h(t)=-16t^2+64t+6 , where t is time in seconds and h(t) is height. How long will it take the T-shirt to reach its maximum height? What is the maximum height?
The T-shirt takes
The T-shirt's maximum height is
To find the time it takes for the T-shirt to reach its maximum height, we need to find the value of t when the derivative of the height function h(t) is equal to 0.
Taking the derivative of the height function gives us:
h'(t) = -32t + 64
Setting h'(t) equal to 0 and solving for t:
-32t + 64 = 0
32t = 64
t = 2
Therefore, it will take the T-shirt 2 seconds to reach its maximum height.
To find the maximum height, we can substitute the value of t into the height function h(t):
h(2) = -16(2)^2 + 64(2) + 6
h(2) = -64 + 128 + 6
h(2) = 70
Therefore, the T-shirt's maximum height is 70 feet.